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Solving high-dimensional optimal stopping problems using deep learning

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  • Sebastian Becker
  • Patrick Cheridito
  • Arnulf Jentzen
  • Timo Welti

Abstract

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

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  • Sebastian Becker & Patrick Cheridito & Arnulf Jentzen & Timo Welti, 2019. "Solving high-dimensional optimal stopping problems using deep learning," Papers 1908.01602, arXiv.org, revised Aug 2021.
    • Handle: RePEc:arx:papers:1908.01602
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    3. Philipp Grohs & Arnulf Jentzen & Diyora Salimova, 2022. "Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms," Partial Differential Equations and Applications, Springer, vol. 3(4), pages 1-41, August.
    4. Yuchao Dong, 2022. "Randomized Optimal Stopping Problem in Continuous time and Reinforcement Learning Algorithm," Papers 2208.02409, arXiv.org, revised Sep 2023.
    5. Vikranth Lokeshwar Dhandapani & Shashi Jain, 2024. "Optimizing Neural Networks for Bermudan Option Pricing: Convergence Acceleration, Future Exposure Evaluation and Interpolation in Counterparty Credit Risk," Papers 2402.15936, arXiv.org.
    6. Sebastian Becker & Patrick Cheridito & Arnulf Jentzen, 2020. "Pricing and Hedging American-Style Options with Deep Learning," JRFM, MDPI, vol. 13(7), pages 1-12, July.
    7. Christian Bayer & Denis Belomestny & Paul Hager & Paolo Pigato & John Schoenmakers, 2020. "Randomized optimal stopping algorithms and their convergence analysis," Papers 2002.00816, arXiv.org.
    8. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    9. Roberto Daluiso & Emanuele Nastasi & Andrea Pallavicini & Giulio Sartorelli, 2020. "Pricing commodity swing options," Papers 2001.08906, arXiv.org.
    10. Xuwei Yang & Anastasis Kratsios & Florian Krach & Matheus Grasselli & Aurelien Lucchi, 2023. "Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing," Papers 2309.04557, arXiv.org.
    11. Ludovic Goudenege & Andrea Molent & Antonino Zanette, 2022. "Computing XVA for American basket derivatives by Machine Learning techniques," Papers 2209.06485, arXiv.org.
    12. Erhan Bayraktar & Qi Feng & Zhaoyu Zhang, 2022. "Deep Signature Algorithm for Multi-dimensional Path-Dependent Options," Papers 2211.11691, arXiv.org, revised Jan 2024.
    13. Nader Karimi & Erfan Salavati & Hirbod Assa & Hojatollah Adibi, 2023. "Sensitivity Analysis of Optimal Commodity Decision Making with Neural Networks: A Case for COVID-19," Mathematics, MDPI, vol. 11(5), pages 1-15, February.
    14. Laurens Van Mieghem & Antonis Papapantoleon & Jonas Papazoglou-Hennig, 2023. "Machine learning for option pricing: an empirical investigation of network architectures," Papers 2307.07657, arXiv.org.
    15. Beatriz Salvador & Cornelis W. Oosterlee & Remco van der Meer, 2020. "Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks," Mathematics, MDPI, vol. 9(1), pages 1-20, December.
    16. Kentaro Hoshisashi & Yuji Yamada, 2023. "Pricing Multi-Asset Bermudan Commodity Options with Stochastic Volatility Using Neural Networks," JRFM, MDPI, vol. 16(3), pages 1-23, March.
    17. Bernard Lapeyre & Jérôme Lelong, 2021. "Neural network regression for Bermudan option pricing," Post-Print hal-02183587, HAL.
    18. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Neural Optimal Stopping Boundary," Papers 2205.04595, arXiv.org, revised May 2023.
    19. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2023. "Deep stochastic optimization in finance," Digital Finance, Springer, vol. 5(1), pages 91-111, March.
    20. Mike Ludkovski, 2020. "mlOSP: Towards a Unified Implementation of Regression Monte Carlo Algorithms," Papers 2012.00729, arXiv.org, revised Oct 2022.
    21. Beatrice Acciaio & Anastasis Kratsios & Gudmund Pammer, 2022. "Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer," Papers 2201.13094, arXiv.org, revised Mar 2023.

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